Download 2.1 The Tangent Ratio

2.1 The Tangent Ratio
In this Unit, we will study Right Angled Triangles. Right angled triangles are triangles
which contain a right angle which measures 90˚ (the little “box” in the corner
means that angle is a right angle).
Skill: Labeling Right Angle Triangles
Quite often, the right triangle will be labeled using letters such as in the triangle above. Capital
letters are used to label __________ while small letters are used to label ____________. We
also need to be able to label the triangle with words and symbols.
The Angle: The size of the one angle we want to use to “start” working with.
Although we will always be given a “right angle”, that will never be the
angle we want to start using as a reference point. It will be one of the
other two angles. This angle will have an actual degree size in the corner or it may have
a little
symbol like this in the corner.
We also use the Greek symbol Ѳ (Theta) in the corner for an angle.
Hypotenuse: This is the side of the triangle directly across from the
right angle (BTW it is always the longest side in a right angle triangle).
Always label this side first.
Opposite: Imagine you are standing in the corner of the triangle
where “the angle” is. The side we call “OPPOSITE” is the side that is
directly opposite you (but not the hypotenuse).
Adjacent: The word adjacent means “beside”. If you are standing
the corner where “the angle” is, the adjacent side is the side “beside you”
(but not the hypotenuse!)
NOTE: Sometimes the Opposite and Adjacent sides are also called LEGS.
in
Example: Label the following triangles using the terms from above.
θ
Foundations and Pre-Calculus Math 10
Chapter 2
Page 1
The Greek scientist and mathematician Ptolemy (90 AD – 168 AD) can be credited with establishing
much of what we are about to learn regarding Trigonometry in this chapter. What he and his fellow
mathematicians found was that if you drew any number of right angle triangles, of any size, certain
patterns emerged. For example, if you were to choose a certain degree of an angle and draw different
large size and small size triangles with that size angle in the corner, and you were to measure the
opposite side and the adjacent side and divide them, you would get the exact same decimal answer for
every single one of these triangles. This is because these triangles are Similar Triangles.
Similar triangles have all three angles the same size and all three sides Proportional.
What followed was a complete list of all the decimal values that you would get if you were to draw every single
sized angle in a right angle triangle and divide the opposite side by the adjacent side. This information became
so important and so well used that it soon developed its own name. It is known as the Tangent Ratio.
TheTangent of Angle A =
Length of theoppositeside
Length of theadjacent side
In simpler terms it usually looks like this:
Tan A =
Opposite
Adjacent
All of these decimal values have been stored in your scientific calculator. Let’s check:
First, make sure your calculator is in the correct “mode”. It needs to say Deg, Degree or D at the top. If
it says G, Grad, R or Rad, your calculator is in the incorrect mode and needs to fixed!
Imagine drawing a triangle with a 60˚ angle in the corner. Imagine measuring opposite and the adjacent
sides and dividing them using the tangent ratio. Do you think your decimal will be larger or smaller than
1? ________ Larger or smaller than 0.5?__________
On your calculator press the following buttons: 60 and then tan. You should see 1.74205. If you don’t
see that, you may need to press the buttons in the order tan and then 60 for your calculator.
We always round off this decimal from our calculator to 4 decimal places. This will be 1.7321
(Remember – if the first number we are “getting rid of” is 5 or bigger, we “bump” the one in front of it
up one value)
This means that every triangle in the world with a 60˚angle in the corner will have an opposite side divided by
an adjacent side that will always turn into 1.7321.
G
Example: Find the Tangent Ratio for the following Triangles.
8
A
22
11
Note: The above question will often be asked another way : Determine tan A and tan G
Foundations and Pre-Calculus Math 10
4
Chapter 2
Page 2
Things to Remember from Last Year:
All three Angles in a Triangle Add to 180˚.
The Pythagorean Theorem:
Example: If tanA = 0.5418, determine the measure of
to the nearest degree.
Our calculator has the decimal value for every size angle and it’s tangent. This question is asking us to
look at the question from the opposite direction – we are given the decimal that came from dividing
the opposite and the adjacent sides. What size was the angle in the triangle?
We are going to have to use the SHIFT or the 2nd button in our calculator along with tan to find
this answer. You are either going to press the buttons 0.5418 then SHIFT/2 nd then Tan Or
you are going to press SHIFT/2nd then Tan then 0.5418.
The calculator shows us _________________ which is the size of the angle with decimals to be
very exact. Our question asks us to round to the nearest degree. Our answer will be ________.
Example: Determine the measures of
R and
S
s
STEPS:
1. Decide which angle to use as your “Starting
Point”.
2. Label your angle, your hypotenuse, your
opposite and your adjacent
18
T
s
S to the nearest tenth of a degree.
R
11
Foundations and Pre-Calculus Math 10
3. Write down the formula
tan A
Opp
Adj
4. Fill in the formula with information from
your triangle. The “A” will become the letter
you chose as your starting point. The Opp
will be the length of the opposite side and
the Adj will be the length of the adjacent
side.
5. Divide your fraction. Leave your answer to 4
decimal places.
6. Use the SHIFT/2nd procedure to find your
angle.
7. Our answer is to tenths which means round
to one decimal!
8. To find the other angle, subtract the angle
you found and the right angle from 180˚
Chapter 2
Page 3
Other Terms to Learn:
Acute Angle: An angle whose size is between 0˚ and 90˚
Angle of Inclination (Also known as Angle of Elevation): The angle
measured between a horizontal line and a line angling upwards.
Example: Ms. C is standing in the teacher parking lot in front of her vehicle which is 32 m away from the
school and directly across from her classroom. She looks up to the second floor and sees the windows of her
classroom which are 16 m high (and is annoyed when she sees her students hanging out of the windows instead
of doing their work). What is the angle of inclination that Ms. C is looking up at? Leave your answer to the
nearest hundredth. Please disregard Ms. C’s height (she’s pretty short anyway).
STEPS:
9. Draw a diagram. Be sure that your
angle of elevation contains a
horizontal line _____ and a line
slanting upwards. Put all given
numerical information on the
diagram.
10. Label your angle, your
hypotenuse, your opposite and
your adjacent
11. Write down the formula
tan A
Opp
Adj
12. Fill in the formula with
information from your triangle.
Divide your fraction. Leave your
answer to 4 decimal places.
13. Use the SHIFT/2nd procedure to
find your angle.
14. Our answer is to hundredths
which means round to two
decimals!
2.1 Assignment: Page 75 #3ac, 4ab, 5ab, 8, 10, 11, 14, 17 BONUS: 21
Foundations and Pre-Calculus Math 10
Chapter 2
Page 4
2.2 Using The Tangent Ratio to
Calculate Lengths
In this section, we will continue to use right angled triangles and the Tangent ratio. This time we will
use the Tangent ratio to find unknown sides in the triangle instead of finding the unknown angle.
Example: Find the length of x to the nearest tenth.
9
22 D
F
x
STEPS:
1. Label all of your given information
using: angle, hypotenuse,
opposite, adjacent.
2. Write down the formula
tan A
Opp
Adj
3. Fill in the formula with
information from your triangle.
4. Use your “Tan” button to find the
Tan of the angle. Do not use your
shift or 2nd key when you know
the number beside the word Tan!
Write down the number from the
calculator to 4 decimals.
5. Put the decimal on the left side
over 1.
6. Cross multiply to find the answer
to your unknown side!
Example: Find the length of RS to the nearest hundredth. (Note: Sides that are named with two
capital letters like RS refer to the side between the angles at R and S)
7
R
D
F
T
39
S
Foundations and Pre-Calculus Math 10
Chapter 2
Page 5
Example: The following picture shows what the current “Casino Regina” looked like when it was
the Union Train Station in Regina in 1911. Let’s assume that the building is 21 m tall. There is a person
sitting in the Model T car that is pointed out with the arrow. The person looks up at an angle of
elevation of 65˚ and sees a Pigeon sitting on the roof. Using trigonometry, determine how far the car is
from the building.
b) If the person in the car wanted to know how far they were from the pigeon along “his line of sight”
how could you find that out? Calculate that distance.
2.2 Assignment: Page 82 #3ac, 4ac,5b, 6, 7, 10, 11, 14
Foundations and Pre-Calculus Math 10
Chapter 2
Page 6
2.4 The Sine and Cosine Ratio &
Calculating Unknown Angles
In this section, we will continue to use right angled triangles and learn two new ratio’s: the Sine Ratio
and the Cosine Ratio. We will use these ratio’s to find unknown angle measures. We use these
formula’s in much the same way we used the Tangent Ratio.
TheCosineof Angle A =
Length of theadjacent side
Length of the hypotenuse
In simpler terms it usually looks like this:
Cos A =
TheSineof Angle A =
Adjacent
Hypotenuse
Length of theoppositeside
Length of the hypotenuse
In simpler terms it usually looks like this:
Sin A =
Opposite
Hypotenuse
Example #1: Determine the sine and cosine of each angle to the nearest hundredth. (Note: when
you are asked to find the sin or cos or tan of an angle it is asking you to use the sin or cos or tan button
along with the angle in your calculator – do NOT use your Shift/2nd button!)
a) 62˚
b) 22˚
Example #2: Determine the measure of each T to the nearest degree. (Note: When you are
asked to find the size of the angle using sin or cos or tan – you will ALWAYS use the Shift/2nd button on
your calculator!)
a) sin T = 0.7834
b) cos T = 0.1279
Foundations and Pre-Calculus Math 10
c) tan T = 1.5834
Chapter 2
Page 7
Example #3: Find the length of the unknown angles
using either sin or cos.
8
T
11
Steps to Finding Unknown Angles:
1. Label all of your given information using:
angle, hypotenuse, opposite, adjacent.
2. Which information do you have numbers
for? Make a list from the choices: Opposite,
Adjacent, Hypotenuse, Angle. Which formula
of the following contains the information you
have?
opp
adj
opp
tan A
, cos A
, sin A
adj
hyp
hyp
3. Fill in the formula with information from your
triangle.
4. Solve the equation you filled in. Remember:
if the variable is beside the word Sin, Cos or
Tan you need to use the Shift or 2nd
button!!!!
5. Subtract the known angles from 180˚ to find
the third angle.
Example #4: Find the measure of each angle to the nearest degree.
A
HINT:
1. Always label your triangle using the words
Angle, Hypotenuse, Opposite and Adjacent.
If you know (or are looking for) the Opposite
and the Hypotenuse, use Sin. If you know (or
are looking for) the Adjacent and the
Hypotenuse, use Cos.
13
C
11
B
Example#5: For the ratio
sin B
2
7
sketch a right triangle that would go along with the ratio.
2.4 Assignment P95 # 4ii, 5ac, 6ac, 7ac, 8ac, 9 ab(just draw one
triangle each), 10ab, 12, 13
Foundations and Pre-Calculus Math 10
Chapter 2
Page 8
2.5 The Sine and Cosine Ratio &
Calculating Unknown Sides
In this section, we will continue to use the Sine (sin) and Cosine (cos) ratios in right angled triangles.
We will use these ratios to find unknown sides in the triangle.
Example #1: Find the length of QS to the nearest
tenth. S
s
Q
s
16
43
R
Steps to Finding Unknown Sides:
1. Label all of your given information using:
angle, hypotenuse, opposite, adjacent.
2. Which information do you have numbers
for? Make a list from the choices: Opposite,
Adjacent, Hypotenuse, Angle. Which formula
of the following contains the information you
have?
opp
adj
opp
tan A
, cos A
, sin A
adj
hyp
hyp
3. Fill in the formula with information from your
triangle.
4. Solve the equation you filled in. Remember:
If there is a NUMBER beside the sin, cos or
tan you do NOT use the shift/2nd button!!!!
5. Make sure you round the length of the side
to the correct number of decimals.
Example #2: Find the length of each side to the nearest tenth. Use sin or cos or tan to find the
first side and use the Pythagorean theorem to find the second side.
G
9
F
Foundations and Pre-Calculus Math 10
Chapter 2
4
E
Page 9
Example #3: Ms.C’s Calculus class was standing outside of the school
for the Halloween class picture. Ms. C looked up at an angle of elevation
of 67˚ and saw a UFO approaching the school. If the UFO is at a height of
231 m, how far is the UFO away from Ms. C along her line of sight?
Example #4: A person leaves the front steps of the parliament
buildings in Regina, walks 35 m North, turns to the right and walks
East for a while and then stops. The angle between his original path
going north and the line that now connects where he stopped and
the front steps of the parliament building is 50˚. How far is he now
from the front steps of the parliament building?
2.5 Assignment P101 #3ac, 4ac, 5ac, 6-9
Foundations and Pre-Calculus Math 10
Chapter 2
Page 10
2.6 Using Trig Ratios to Solve
Right Triangles
In this section, we will use Sin, Cos and Tan to Solve Right Triangles. Solving a right triangle
means that at the end of each question you will know the length of all three sides and the size of all
three angles. We can use the following to help us solve the triangles:
SinA =
opp
adj
opp
, Cos A=
, Tan A=
,
hyp
hyp
adj
a2
b2
Example #1: Solve the following triangles. Find side
lengths to the nearest tenth and angles to the nearest degree.
a)
Foundations and Pre-Calculus Math 10
c2 ,
A
B
C 180
Steps to Solving Triangles:
1. Label all of your given information using:
angle, hypotenuse, opposite, adjacent.
2. Which information do you have numbers
for? Make a list from the choices: Opposite,
Adjacent, Hypotenuse, Angle. Which piece of
information do you want to find first? Which
formula of the following contains the
information you have?
opp
adj
opp
tan A
, cos A
, sin A
adj
hyp
hyp
3. Fill in the formula with information from your
triangle.
4. Solve the equation you filled in. Remember:
If there is a NUMBER beside the sin, cos or
tan you do NOT use the shift/2nd button!!!!
5. Make sure you round the length of the side
or angle size to the correct number of
decimals.
6. To find the THIRD side: Use the
Pythagorean theorm.
7. To find the THIRD angle: Use the 180
rule.
8. Circle all the side lengths and angle sizes that
you found.
9. Redraw your original triangle and fill in all
three angle sizes and all three side lengths.
Chapter 2
Page 11
b)
c)
2.6 Assignment P111 #6, 7, 8, 11, 12a
Foundations and Pre-Calculus Math 10
Chapter 2
Page 12
2.7 Solving Problems With more
than one Triangle
In this section, we will use Sin, Cos and Tan to find an unknown side or angle in situations where more
than one triangle is “attached together” in a problem.
Example #1: Find the length of BC to the nearest
tenth.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Foundations and Pre-Calculus Math 10
Steps to Solving Multiple
Triangle Problems:
Draw a diagram!
Find a triangle where you can see two pieces
of information in it.
Label all of your given information in this
triangle using: angle, hypotenuse, opposite,
adjacent.
You need to choose which additional piece of
information to find. Usually you will be
looking for a side that is “shared” with the
other triangle you haven’t used yet. This will
be your “??” .
Make your “list” on the side with the two
known numbers and your ??.
Choose your formula to find “??”
opp
adj
opp
tan A
, cos A
, sin A
adj
hyp
hyp
Fill in the formula with information from your
triangle.
Solve the equation you filled in. Remember:
If there is a NUMBER beside the sin, cos or
tan you do NOT use the shift/2nd button!!!!
Put your answer on your diagram.
Now – look at the OTHER TRIANGLE. Repeat
steps 2 -6 to find the side in the other
triangle that will be your final answer. You
may also need to use the Pythagorean
Theorem or the 180 rule.
Chapter 2
Page 13
Example #2:
2.7 Assignment #1: P118 3ac, 4ac, 5ac
Foundations and Pre-Calculus Math 10
Chapter 2
Page 14
2.7 Continued…….
Example #3:
Foundations and Pre-Calculus Math 10
Chapter 2
Page 15
Example #4:
Foundations and Pre-Calculus Math 10
Chapter 2
Page 16
Example #4:
Foundations and Pre-Calculus Math 10
Chapter 2
Page 17
Example #4:
2.7 Assignment #2: P118 #6, 8, 9, 11, 13, 14
Foundations and Pre-Calculus Math 10
Chapter 2
Page 18
Chapter 2 Helpful Hints Sheet
Steps to Finding Unknown Angles:
1. Label all of your given information using:
angle, hypotenuse, opposite, adjacent.
2. Which information do you have numbers
for? Make a list from the choices: Opposite,
Adjacent, Hypotenuse, Angle. Which formula
of the following contains the information you
have?
opp
adj
opp
tan A
, cos A
, sin A
adj
hyp
hyp
3. Fill in the formula with information from your
triangle.
4. Solve the equation you filled in. Remember:
if the variable is beside the word Sin, Cos or
Tan you need to use the Shift or 2nd
button!!!!
5. Subtract the known angles from 180˚ to find
the third angle.
Steps to Finding Unknown Sides:
1. Label all of your given information using:
angle, hypotenuse, opposite, adjacent.
2. Which information do you have numbers
for? Make a list from the choices: Opposite,
Adjacent, Hypotenuse, Angle. Which formula
of the following contains the information you
have?
opp
adj
opp
tan A
, cos A
, sin A
adj
hyp
hyp
3. Fill in the formula with information from your
triangle.
4. Solve the equation you filled in. Remember:
If there is a NUMBER beside the sin, cos or
tan you do NOT use the shift/2nd button!!!!
5. Make sure you round the length of the side
to the correct number of decimals.
Foundations and Pre-Calculus Math 10
The Pythagorean Theorem:
Steps to Solving Triangles:
1. Label all of your given information using:
angle, hypotenuse, opposite, adjacent.
2. Which information do you have numbers
for? Make a list from the choices: Opposite,
Adjacent, Hypotenuse, Angle. Which piece of
information do you want to find first? Which
formula of the following contains the
information you have?
opp
adj
opp
tan A
, cos A
, sin A
adj
hyp
hyp
3. Fill in the formula with information from your
triangle.
4. Solve the equation you filled in. Remember:
If there is a NUMBER beside the sin, cos or
tan you do NOT use the shift/2nd button!!!!
5. Make sure you round the length of the side
or angle size to the correct number of
decimals.
6. To find the THIRD side: Use the
Pythagorean theorm.
7. To find the THIRD angle: Use the 180
rule.
8. Circle all the side lengths and angle sizes that
you found.
9. Redraw your original triangle and fill in all
three angle sizes and all three side lengths.
Chapter 2
Page 19